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First published online March 14, 2005
Journal of Experimental Biology 208, 1125-1146 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01507
Jet flow in steadily swimming adult squid
Department of Applied Ocean Physics and Engineering, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 02543, USA
* Author for correspondence (e-mail: mgrosenbaugh{at}whoi.edu)
Accepted 24 January 2005
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Summary |
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 TOP Summary IntroductionMaterials and methodsResultsDiscussionReferences |
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Key words: swimming, squid, Loligo pealei, jet propulsion, hydrodynamics, DPIV
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Introduction |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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;
Lighthill, 1969;
Hoar et al., 1994;
Anderson and DeMont, 2005). At
low speeds and in hovering, L. pealei use both fin and jet
propulsion. As swimming speed increases, the contribution of the fins to the
production of thrust decreases, fin gait changes, and squid rely more heavily
on jet propulsion. At high speeds the fins are often wrapped tightly against
the mantle (Williamson, 1965).
Squid attain some of their most dramatic speeds and accelerations during
escape responses, known as escape jetting
(Gosline and DeMont, 1985).
During such movements squid rely almost entirely on jet propulsion. In
addition to its role in thrust production, the jet is the primary exhalant
pathway for respiration (Gosline et al.,
1983). In this paper, we focus on characteristics of the squid jet
in steady swimming.
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For more than 40 years, two conflicting models have been used to analyze
the hydrodynamics of squid jet propulsion: the `squirt', or prolonged jet
model (Trueman and Packard,
1968; Johnson et al.,
1972; O'Dor, 1988;
Anderson, 1998;
Anderson and DeMont, 2000), and
the `puff', or vortex ring model (Seikmann, 1963;
Weihs, 1977). The prolonged
jet model assumes the jet to be an elongated mass of high speed fluid. In a
real fluid, such jets are marked by a three-dimensional shear layer, through
which fluid velocities vary continuously from the velocity of the jet core to
that of the surrounding fluid. The shear layer is unstable, which leads to the
growth of waves. These waves are the seed points for short-lived vortices
whose energy is eventually dissipated into the surrounding flow
(Pai, 1954;
Drazin and Reid, 1981;
Van Dyke, 1982). The rate and
character of these developments is dependent upon jet velocity, orifice
diameter, fluid viscosity, density and local perturbations in the flow. By
contrast, vortex ring propulsion is characterized by the periodic shedding of
individual torroidal fluid structures at the trailing edge of the jet nozzle.
These torroidal flows, or vortex rings, appear in cross-section as two
counter-rotating vortices. Flow at the center of an emitted vortex ring is in
the direction of the original emission of fluid from the jet nozzle, unless
some other forcing causes the ring to rotate on one of its radial axes. Weihs
(1977) demonstrated the
availability of impressive hydrodynamic benefits in properly tuned periodic
jet propulsion by vortex rings. Nevertheless, the actual fluid structure and
velocities in the jets of steadily swimming adult squid have remained largely
unknown.
Recently, Anderson (1998),
Anderson and DeMont (2000), and
Anderson et al. (2001c)
performed both quasi-steady and unsteady analyses of squid hydrodynamics using
highly accurate kinematic data and whole-body deformation extracted from
high-resolution, high-speed video records. Their data revealed that steadily
swimming adult L. pealei emit relatively large volumes of fluid from
a small opening, suggesting the prolonged jet model. At the same time, works
focusing on the hydrodynamics of vortex ring formation
(Gharib et al., 1998;
Linden and Turner, 2001),
theorized that jet-propelled organisms, including squid
(Linden and Turner, 2001),
might use vortex ring propulsion to enhance efficiency. Gharib et al.
(1998) observed that the
formation of vortex rings in jets emitted from cylindrical pipes into still
water was dependent upon the ratio of the length of the plug of fluid expelled
from the pipe, L, and the inside diameter of the pipe, D.
They found that a solitary vortex ring was formed when L/D
3.6-4.5. Fig. 2A shows a
vorticity contour plot from our experimental repetition of this result
(L/D=4.3). Gharib et al.
(1998) observed that all the
vorticity shed from the pipe was bound up in this single vortex ring and that
no other flow structure was present (Fig.
2A). When L/D was greater than about 3.6-4.5 the
vorticity shed from the pipe no longer rolled up completely into a single
vortex ring. Rather, a leading ring formed and a trail of vorticity followed
behind. Fig. 2B shows our
repetition of this result for the case of L/D=16. Linden and
Turner (2001) used theoretical
arguments to arrive at a similar conclusion regarding vortex ring production
and the ratio L/D. Most significantly, their analysis
predicts that vortex rings produced using the highest possible
L/D ratio for the formation of a solitary vortex ring are
characterized by the highest ratio of thrust to jet plug kinetic energy. This
suggests highest propulsive efficiency. In addition, Krueger and Gharib
(2003) showed that vortex
rings exhibit a higher average thrust than that predicted by jet plug momentum
and attributed the phenomenon to an increase in local pressure above ambient
(so-called `over-pressure'), at the jet orifice, due to the formation of the
vortex ring. Even before these findings, Gharib et al.
(1998) suggested that `the
mere existence of the formation number...hints at the possibility that nature
uses this time scale...', e.g. in propulsion.
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Preliminary investigations of jet volume flow and wake structure in
steadily swimming adult squid, however, have not suggested that their
locomotive structure and behavior take advantage of the potential benefits of
L/D ratios near 4. On the contrary, Anderson et al.
(2001a) reported
L/D ratios as large as 40 in steadily swimming adult L.
pealei. Furthermore, the data of Anderson
(1998) and Anderson and DeMont
(2000) can be used to determine
L/D ratios of 39.2-67.2, also in steadily swimming L.
pealei. Bartol et al.
(2001b) reported
L/D ratios of 3-17 in small L. brevis
(Lm<3.0 cm) and 10-40 in large and intermediate-sized
individuals. They claim that benefits from periodic vortex ring propulsion as
described by Weihs (1977)
might be obtained in smaller squid. Jets visualized by dye injection from two
small squid (mean Lm=4.2 cm) swimming more or less
steadily were reported to have left several vortex rings in the wake
(Bartol et al., 2001b). Some
highly turbulent jets were emitted from these squid with no vortex rings
present, but this occurred when the squid swam erratically, apparently in
response to irritation caused by dye injection. In addition, Bartol et al.
(2001b) hypothesize that
larger squid, `...probably produce vortex rings that are too widely spaced
to benefit significantly from ring interaction.' Although our data do not
suggest that adult L. pealei use periodic vortex ring propulsion, jet
frequencies and swimming speeds from Anderson and DeMont
(2000) and the present work
similarly predict wide spacing (>40 cm) between sequentially emitted jet
structures in adult L. pealei.
Despite such observations and predictions about squid jet structure
relative to the ratio L/D, the pipe experiments of Gharib et
al. (1998), from which the
importance of the ratio was revealed, differ in a significant way from jetting
in swimming squid: they were conducted in still water. By contrast, the jets
of squid issue into a background flow past the jet nozzle due to the motion of
the squid through the water. Thus, the question arises as to whether or not a
simple L/D ratio is sufficient to predict squid jet
structure. The significance of this background flow and the failure of
L/D ratios from experiments in still water to predict
structure in this case were pointed out by Anderson et al.
(2001a) in a preliminary
analysis using the squid and pipe jet data presented here. They reported that
vorticity shed into the wake from the outer boundary layer apparently
contributed to a change in jet structure. The sign of the outer boundary layer
vorticity is opposite to that of the jet vorticity, and its magnitude is
dependent on swimming speed, body shape and angle of attack. Consider the
extreme case when background flow velocity is greater than jet velocity.
Intuitively, one realizes that a vortex ring, such as in
Fig. 2A might never form,
regardless of L/D, since the outer boundary layer vorticity
would likely dominate the downstream flow development. In response to this
finding, Jiang and Grosenbaugh
(2002) numerically simulated
jets from pipes in the presence of background flow and confirmed the
significance of background flow on jet structure, as observed in our squid and
pipe jet experiments. Specifically, Jiang and Grosenbaugh
(2002) report a decrease in
the L/D ratio necessary to produce a single vortex ring puff
as background flow is increased. Work by Krueger et al.
(2002,
2003) independently confirmed
this finding, with some slight differences due to their use of different
initial conditions.
In this paper, we present the results of flow visualization from (1) a large number of jets emitted from steadily swimming adult squid, and (2) pipe jet experiments with a background flow parallel to the jet. For squid, we use flow field data to investigate not only jet structure, but also trends in parameters such as jet velocity, slip, jet frequency, jet angle, thrust and propulsive efficiency as functions of swimming speed.
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Materials and methods |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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Swimming conditions
A total of 116 squid jets were recorded in 42 swimming sequences in which
squid were holding station in the flow. The number of consecutive jets per
sequence ranged from 1 to 14. The squid swam in a test section 80 cm long and
23 cm deep. The width of the test section was varied when necessary to
increase the time spent by the squid in the plane of the laser sheet that was
used to illuminate the flow. Three different test section widths were used: 15
cm (86 jets), 35 cm (25 jets) and 40 cm (5 jets). The test section was
constructed in a large, recirculating, open-channel flume (30 cm deep and 78
cmwide) capable of speeds up to 62 cm s-1 and temperature control
to within ±0.1°C. Flow speeds, U, in the test section were
determined from the same DPIV records that were used for jet analysis and
ranged from 10.1 to 59.3 cm s-1, i.e. 0.33 to 2.06
Lm s-1. Water temperature in the flume and
holding tanks ranged from 12.5 to 19.5°C, corresponding to daily
temperatures in the waters of Nantucket Sound. A honeycomb flow-through
barrier was placed at the upstream end of the test section to damp out
large-scale flow disturbances. The barrier was 12.7 cm in streamwise length
with a tube diameter of 1.3 cm. A plastic grid bound the downstream end of the
test section (grid size, 1.5 cm; bar width, 0.2 cm; streamwise length, 0.8
cm). Laser Doppler anemometry from a previous experiment in the same flume
revealed turbulence intensities of 4-6% over the range of experimental flow
speeds (Anderson et al.,
2001b). These fluctuations in flow velocity are significantly
lower than those produced by the presence and jetting of the squid.
Flow visualization
Flow around the swimming squid was visualized by digital particle imaging
velocimetry, DPIV (Adrian,
1991; Willert and Gharib,
1991). DPIV uses cross-correlation to determine the average
displacement of groups of seeding particles in consecutive stroboscopic images
of a two-dimensional slice of a flow field. The particles are grouped in
sub-windows of the field of view by a user-defined grid. A pulsed laser sheet
is used to illuminate the flow, and is synchronized with a digital video
camera so that it flashes once in each video frame. The camera is pointed at
right angles to the laser sheet. The velocity field of the flow is determined
by dividing average local displacements of seeding particles in the flow by
the time step, 
t, between the laser pulses. In this work, we
used a hybrid DPIV code that performs a quick FFT estimate of sub-window
displacements and then refines the estimate to sub-pixel accuracy by local
cross-correlation (McKenna and McGillis,
2002). Sub-windows were 32x32 pixels with an overlap of 16
pixels.
To capture the jets of swimming squid, which are emitted from the ventral
side near the trailing end (Fig.
1), we oriented our laser sheet (New Wave Research, Fremont, CA,
USA; Nd:YAG) streamwise and vertical by one of two methods: (1) directing the
laser sheet up through the transparent bottom of the flume, or (2) deflecting
the beam upstream with a thin (3 cm x 40 cm), vertical mirror at an
angle of 45° to the flow, submerged at the downstream end of the test
section. In the latter arrangement, data was only acquired when the squid
trailing edge was >1 body length upstream of the mirror to minimize
artifacts caused by its presence in the flow. The two arrangements were used
to optimize resolution at different fields of view. A single side-view camera
(Megaplus ES 1.0, Roper Scientific, Vianen, The Netherlands; 1008x1018
pixels) was used to visualize fields of view ranging from 25 to 30 cm on a
side. The laser sheet was 1-2 mm thick and the flow was seeded with
silver-coated, hollow glass spheres, 10 µm in diameter (DANTEC, Skovlunde,
Denmark). The time step, t, between laser pulses was set at
2-10 ms, depending on swimming speed to optimize particle displacements
(McKenna and McGillis, 2002).
The laser pulse length, or exposure time, was 3-5 ns with an available power
of 500 mJ per pulse. However, power was attenuated significantly to minimized
irritation to the animal. The camera was operated at 30 Hz. Therefore, pairs
of exposures, or image pairs, were acquired at 15 Hz. Up to 700 sequential
images were acquired per swimming sequence.
It should be mentioned here that DPIV has some limitations when calculating
flow velocities near bodies, such as squid and pipes. The basic DPIV
algorithm, i.e. cross-correlation, is not able to distinguish between
particles in a flow and patterns on the surfaces of bodies. If the sub-window
being analyzed includes seeded flow and a body surface moving relative to each
other, the patterned region with the higher combination of brightness and
amount of bright area will dominate the velocity calculation in that region.
The deformation of the flow due to shear near the body also affects the
velocity calculation since it alters the pattern of the seeding particles.
Usually, the algorithm results in a velocity somewhere between that of the
body surface and that of the flow. Theoretical and experimental fluid dynamics
tell us that there is, indeed, a continuous variation in fluid velocity from
zero at the body surface with respect to the body to the background flow
velocity at a certain distance from the surface (which is relatively monotonic
in the general case). Therefore, although the magnitudes of velocity and
vorticity calculated in this shear layer are biased, the sign of the
vorticity, and upper-bound values of boundary layer thicknesses, may be
trusted. We have certainly not attempted to make any quantitative conclusions
from the DPIV data where affected by body surfaces, but vorticity contours and
velocities along the squid and pipe surfaces are left in our plots for
instructive reasons (e.g. Fig.
3). Due to the resolution used in our experiments velocities at
points greater than 0.34 cm away from a body surface and vorticities greater
than 1.0 cm away are calculated from sub-windows that do not include the body.
An example of a clear artifact due to the presence of a body can be seen in
Fig. 2B. Note the closure of
the parallel vorticity contours of same sign near the pipe orifice. Fluid
theory states that during jetting the vorticity contours actually originate on
the surfaces of the pipe and piston. Numerical simulations of pipe jets
demonstrate this fact (Jiang and
Grosenbaugh, 2002). However, since the pipe is stationary, zero
velocities are calculated by DPIV over the majority of the area within the
silhouette of the pipe. This results in zero vorticities over much of the same
region, and the contour plotting program closes the non-zero jet vorticity
contours outside this region of the plot.
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Squid jet analysis
The fluid structure of squid jets was analyzed qualitatively using velocity
field plots and vorticity contour plots, but we were also able to digitize the
region of the jet and extract quantitative information such as jet length,
width, angle and average jet velocities. Preliminary inspection of the video
records and velocity fields around swimming squid showed that in every case,
average jet velocity was greater than the background velocity. Therefore, the
structure of the jet was digitized by taking the region within which flow
speeds were at least 5% greater than the maximum background flow speed
upstream of the jet and away from the squid. 5% was used because, in general,
it appeared to be the lowest threshold that correctly filtered out velocities
due to the variation in the flume flow, while correctly digitizing the
jet.
Since jet structure was observed to be relatively elongated and straight,
with a central core made up of the highest velocity fluid, a jet centerline
was determined to aid in the analysis. A rough centerline was drawn manually
in MATLAB for each visualization (i.e. each processed image pair) that
exhibited a jet flow. The centerline line was refined iteratively by
calculating a best-fit line through the centroids of velocity calculated in
slices of the jet perpendicular to the previously estimated centerline,
starting with the rough centerline. For three-dimensional approximations, we
treated the slices as disks. We call the disk diameter, `the diameter of the
jet fluid structure', or more simply, `jet structure diameter',
Dj. Length of the visualized jet fluid structure,
Lj, and jet angle, ß, were also determined using the
jet centerline. Jet angle was determined early in the jet period to avoid
artifacts resulting from the deformation of the jet in its later development.
A proper jet angle measurement was possible in 110 of the 116 jets. In
addition to the centerline, we determined a curve for each jet representing
the position of maximum velocity in the slices perpendicular to the refined
centerline. In general, this curve followed the centerline closely. We refer
to the velocity along this curve as jet core velocity,
ujc. The average jet core velocity,
jc, is defined as the spatial average velocity
along this curve.
We calculated average jet velocity in three ways to generate (1) a standard
value, j, (2) an upper-bound value,
jH, and (3) a lower-bound value,
jL, for each jet. The standard value was calculated
by taking the time average of the average jet core velocity,
jc, during a given jet period. The upper bound was
calculated by taking the time average of the maximum jet core velocity. The
lower-bound value was calculated by taking the time average of the average jet
velocity over the entire jet structure, taken in slices of the jet
perpendicular to the centerline. The value for each slice was weighted by
slice volume, assuming the slices to be disks of diameter
Dj to account roughly for three-dimensionality. Although
this is clearly an approximation, it results in a lower estimate than a simple
area-based average of the jet cross-section, since slices where
Dj was larger tended to have lower velocities. A fully
axi-symmetric rendering of the jets was not used for the lower-bound
calculation because the jet velocities were not always sufficiently symmetric
about the centerline or the jet core curve. The actual average jet velocity is
expected to lie somewhere between the standard value,
j, and the lower-bound value,
jL. It should also be noted that our jet velocities
have likely been decreased by deceleration and entrainment, especially the
lower-bound value, as the jet fluid enters the surrounding fluid. With that in
mind, and assuming the jet profile at the jet orifice to be relatively
undeveloped due to the short, funnel-shaped nozzle, we use the standard value
as the best estimate of actual average jet velocity. Therefore, unless
otherwise specified, the term average jet velocity, when used in reference to
our data, refers to the standard value, j. The term
jet velocity, uj, will be used here to represent
instantaneous jet velocity assuming a constant velocity profile at the jet
orifice.
Two measurements of jet length were used to compare the data from all squid jets observed. We call the first `jet structure length', Lj. It is defined as the maximum length observed for the digitized fluid structure of the jet during the jet period. This measurement frequently underestimated the actual length of the jet structure, because (1) the jets of squid often extended beyond of the field of view and (2) the squid were often slightly `yawed' during jetting. Yaw is the angle between the oncoming flow and the dorso-ventral plane of the animal. We determined yaw qualitatively from the side-view images, as evidenced by unequal lighting by the laser sheet along the squid body. Yaw resulted in jets that passed through the laser sheet, and the cross-section of a jet thus visualized is almost without exception shorter than the jet itself.
The second measurement of jet length used was `jet plug length',
L, which was determined by multiplying the average jet velocity by
the jet period, tj. Upper and lower bounds for
L were also determined using the upper and lower-bound average jet
velocities. We define jet period here, as the time over which the jet orifice
was observed to be fully open as determined from the video records. In
general, this period tended to be the majority of the time between orifice
opening and closing. Anderson and DeMont
(2000) observed the same
behavior and included a detailed plot of jet diameter as a function of time.
Of the 116 jets that were recorded, the jet nozzle was visible in 89. We
report jet structure length Lj and jet plug length
L in proportion to the average jet orifice diameter D of 0.8
cm during the jet period, as measured by Anderson and DeMont
(2000) on similarly sized
L. pealei. This allows our squid jet data to be viewed in comparison
to the pipe jet experiments of Gharib et al.
(1998).
Jet frequency f and average jet frequencies favg were determined from our video records. The frame numbers of the images in which the jet nozzle first opened were recorded when possible. The number of frames between consecutive jet openings was divided by the frame rate of the camera (30 Hz) to obtain the elapsed time between the jets. This time period is called the `locomotive period' rather than the `jet period' because it represents a full locomotive cycle, i.e. jetting and refilling. Locomotive period was then inverted, yielding jet frequency, f. There were 61 full locomotive periods in which the jet nozzle was visible. Jet frequency was calculated for each of the 61 periods. From these 61 values, favg was determined for each set of all locomotive cycles representing the same squid and swimming speed. Only sets with at least three locomotive cycles were used. There were 9 such sets spanning the full range of swimming speeds observed.
Propulsive efficiency
Propulsive efficiency is the hydrodynamic efficiency during propulsion.
Anderson and DeMont (2000)
found that the equation they derived for the hydrodynamic efficiency of squid
during jetting was the same equation used in fluid dynamics to determine
propulsive efficiency in rockets (Streeter
and Wylie, 1985; Houghton and
Carpenter, 1993). In addition, they derived an equation for the
hydrodynamic efficiency in squid for the whole locomotive cycle. Anderson and
DeMont (2000) explain that
these two equations are the appropriate efficiency equations for squid, rather
than the Froude efficiency equation, since Froude efficiency assumes a
constant forward intake of the working fluid while squid use an aft-facing
intake system (Fig. 1). In this
investigation, we have extended the hydrodynamic efficiency equations
prescribed for squid by Anderson and DeMont
(2000) to include jet angle.
Anderson and DeMont (2000)
incorporated jet angle into propulsive efficiency by simply using the axial
component of jet velocity for jet velocity, uj, in their
equations. This approach, however, does not correctly account for the decrease
in efficiency expected due to a non-zero jet angle.
The derivation of hydrodynamic efficiency, 
, begins with the equation,
 | (1) |
Fluid theorists have defined the rate of useful work as thrust T
multiplied by forward velocity, thus TU
(Prandtl, 1952;
Streeter and Wylie, 1985;
Houghton and Carpenter, 1993).
This is applied both to stationary propellers with flow past and to propellers
translating at constant velocity. Wasted energy is defined as any kinetic
energy left in the wake as a result of jetting relative to the surrounding
flow, that is, the kinetic energy of the jet signature. This is best
understood considering an astronaut who is propelled by throwing a wrench
vs pushing off an immovable object using the same force for the same
amount of time. In both cases the astronaut is accelerated to the same speed,
but in the former case the astronaut does more work because the wrench gives
way, or `slips'. That is, the force acts over an additional distance. This is
analogous to a fluid dynamic concept known as `slip'. Slip is simply the
fraction by which jet velocity exceeds or falls short of the surrounding flow
speed and is frequently used as an indicator of propulsive efficiency in an
inverse sense. High slip indicates low efficiency. It is defined as
uj/U-1, where uj is the jet
velocity, assuming uniform, ideal flow, and U is the velocity of the
surrounding fluid, both with respect to the jet orifice. Slip is traditionally
defined as one-dimensional, i.e. with uj and U in
the same direction. Nevertheless, it is obvious that a jet issued at an angle
ß to a uniform flow has greater `slip' than
uj/U-1. If the angle between U and
uj were increased to 90°, slip would be infinite.
Therefore one might argue that the definition of slip could be expanded to
include a jet angle by substituting the component of U in the
direction of uj for U in the equation of slip,
i.e. slip=uj/(Ucosß)-1. In this paper, we
calculated slip using this equation and the one-dimensional form, for
comparison.
If uj is in the same direction as U, the rate
at which excess kinetic energy is added to the surrounding fluid is
Q(uj-U)2/2, where
is the jet fluid density, and Q is the rate of volume flow out of the
jet. For ideal, uniform jet flow, thrust is Quj and
therefore the rate of useful work, TU, is
QujU. Substituting these into Eq. 1, one
arrives at the equation for rocket motor propulsive efficiency,
r:
 | (2) |
If instead we define an angle ß between uj and
U, excess kinetic energy becomes
Q[(ujcosß-U)2+(ujsinß)2]/2
and useful work becomes
Q(ujcosß)U. Substituting these
into Eq. 1, we find that the propulsive efficiency of a jet,
j, issuing at an angle ß to U is:
 | (3) |
Since we define jet angle as the angle between uj and
U, any useful work done by the vertical component of the squid jet
during the jet period can be incorporated if the vertical motion of the body
during jetting is known. U and ß can then be determined from the
resultant of the horizontal and vertical body velocities. Anderson
(1998) observed small upward
movements (<1 cm) in steadily swimming adult L. pealei during
jetting apparently due, in part, to the vertical component of the jet. When
these motions are accounted for, TU increases, jet angle decreases
and wasted kinetic energy decreases, therefore propulsive efficiency
increases. Nevertheless, the contribution is expected to be very small,
especially at high swimming speeds, where the horizontal swimming speed
dominates the motion of the squid. Anderson
(1998) reports a vertical speed
of about 1.6 cm s-1 during the jet period of a squid swimming with
a horizontal speed of 25 cm s-1 (1.0 Lm
s-1) and a jet angle of 30°. Taking this vertical motion into
account, we calculate an increase in propulsive efficiency during the jet
period of just 2%.
If we begin again with Eq. 1, treat the work required for mantle refill as
wasted energy and assume that the same volume of fluid is taken in as jetted
out in any cycle, we obtain the whole-cycle hydrodynamic efficiency for squid,
wc:
 | (4) |
where uR is the refill velocity relative to the body at
the intake orifice. As in the derivation of the whole-cycle efficiency
equation of Anderson and DeMont
(2000), the assumption that the
total volume emitted during jetting is equal to the total volume taken in
during refill results in the convenient elimination of jet period
tj and refill period tR
from the equation. Once again, a simple multiplier of cosß in the
numerator turns out to be the only difference between the corresponding
one-dimensional equation derived by Anderson and DeMont
(2000). Anderson and DeMont's
whole-cycle hydrodynamic efficiency predicts a theoretical limit of 58%, due
largely to the cost of mantle refill. This limit is approached when
uj=1.7U and uR is small. The
same is true for Eq. 4, except that the theoretical limit is 58% x
cosß. Both Eqs. 3 and 4 demonstrate clearly and correctly that propulsive
efficiency decreases to 0 as jet angle increases from 0° to 90°.
At the present time, no accurate measurements of refill velocities have
been achieved. Mantle refill volume flows have been measured
(Anderson and DeMont, 2000),
but refill orifice area is unknown. Therefore refill velocities must be
estimated to calculate whole-cycle efficiencies. We have estimated the average
refill area to be approximately 2-3 times the jet orifice area, based on rough
visual estimates. Refill periods tR, in general,
are about 1-2 times the jet period tj, based on our
data and that of Anderson and DeMont
(2000). Therefore, assuming
that the average volume outflow during jets is equal to average volume intake
during refill in steady swimming, we estimated refill velocities to be
0.17-0.5 times jet velocity and used this range in calculations of whole-cycle
propulsive efficiency.
It should be noted here that Eqs. 2 and 3 produce the most accurate values
of efficiency when instantaneous values of uj, U
and ß are available. An accurate average propulsive efficiency during the
jet period can then be determined from the time average of the instantaneous
efficiency over the jet period. This is for the simple reason that the time
average of 
, for example, is not the
same as unless
uj is constant. The same is true in division,
multiplication, cosine and other non-linear mathematical functions. By
contrast, it is incorrect to use instantaneous values in Eq. 4 since the
equation represents the whole cycle as a unit, therefore time averaged values
for uj, uR, U and ß
during their respective periods must be used. If the instantaneous values
spend most of the time far below and far above their time averages, it is
better to return to first principles (Eq. 1) and rigorously determine
efficiency from useful work and wasted kinetic energy. Nevertheless, we and
Anderson and DeMont (2000) have
observed that steadily swimming adult squid maintain uj,
U and ß with relatively small variation about their time
averaged values for the majority of their respective periods, and Eqs. 2, 3,
and 4 are expected to give meaningful efficiencies even using time averaged
values. As a test of Eq. 2, total kinetic energy, KET, and
excess kinetic energy, KEe, of the jet fluid were
calculated from a three-dimensional approximation of jet velocity from all jet
visualizations. These were used to calculate propulsive efficiency by
(KET-KEe)/KET.
Pipe jet experiments
Jets emitted from a cylindrical pipe were visualized using DPIV. Fluid was
driven out of the pipe by a motor-actuated piston. The inside and outside
diameters of the pipe were 2.39 and 2.54 cm, respectively. The trailing edge
of the pipe was beveled to about 30° to form a sharp edge with a diameter
equal to the inside diameter, D. The pipe was aligned parallel to the
bulk flow in the flume (i.e. streamwise) and centered between the top, bottom
and side walls. The free surface was eliminated with a sheet of acrylic. Total
water depth was 22 cm and the side walls were 78 cm apart. The pipe was
mounted so that the outer flow over the last 1.2 m section of pipe encountered
no mounting structures to interrupt the flow. A faired beam coupled the piston
to its motor 3 m upstream of the jet nozzle. In general, the piston program
consisted of a rapid acceleration, followed by constant speed and a rapid
deceleration, similar to the time course of jet velocity
uj observed in L. pealei
(Anderson and DeMont, 2000).
Piston velocities up, i.e. average jet velocities, ranging
from 1 to 10 cm s-1 and background flow speeds U of 0-15
cm s-1 were used. Ratios of jet plug length to jet orifice
diameter, L/D, ranging from 2 to 16 were examined.
Experiments in which jets were emitted into still water (U=0) were
performed to show that the apparatus gave results that matched those of Gharib
et al. (1998)
(Fig. 2).
The purpose of our pipe experiments was to examine the effect of background
flow on jet structure and to attempt to mimic the fluid structures observed in
the jets of squid. When squid jet, the nozzle opens suddenly on the dorsal
side of the body and fluid is emitted at various angles to the horizontal. The
jet is emitted into the surrounding flow, not into the wake of the squid, and
is affected by a boundary layer that begins peeling off the trailing edge of
the jet nozzle the moment it opens. With this in mind, we developed a
technique that we call `pre-jetting' for use in our pipe jet experiments. In
pre-jetting, we moved the piston at the same speed as the background flow
until the flow behind the pipe was much more similar to the background flow
than to the wake of the pipe while not jetting. This was determined from
preliminary visualizations of the flow. Immediately following pre-jetting, the
piston program for the jet was started. We do not assert that our pipe and
squid jet initial conditions are identical. Pre-jetting certainly does not
eliminate the vorticity shed from the inner and outer boundary layers of the
pipe, and therefore the flow into which the jet was emitted was not entirely
uniform. Also, the flow past the jet nozzle of the squid is certainly not
uniform about the circumference of the nozzle, due to the close proximity of
the nozzle to the head and arms (Fig.
1). Nevertheless, in both cases, jets are emitted into a
background flow, not a wake, and they are impacted by boundary layers
separating from the trailing edge of the jet nozzle. Furthermore, the fluid
structure of pipe jets modulated in this way most closely resembled squid
jets. Two other initial conditions were attempted in pipe jets: (1) starting
the piston from rest and emitting the jet into the wake of the pipe, and (2)
sucking the wake into the pipe before jetting. Both situations resulted in
vortex patterns that were, in general, qualitatively less similar to those of
squid jets. Krueger et al.
(2003) used different initial
conditions in which both the background flow and the jet flow start from rest.
Although this may mimic a jet-propelled organism starting from rest, it is
different from the initial conditions for a jet emitted from a steadily
swimming squid. Some earlier examples of visualizations of jets in the
presence of background flow are displayed by Yamashita et al.
(1996).
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Results |
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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Velocity vectors were left out of Figs 3, 4, 5, 6 since the plots were intended to demonstrate the basic structure of the jet. Furthermore, the jet velocity was generally parallel to the axis of the elongated jet structure. A representative jet is shown in more detail and with vectors in Figs 7 and 8. Fig. 7A shows the velocity field from Fig. 3A at a higher resolution and with velocity vectors. Even at this resolution every other vector has been removed for clarity. The jet itself, extracted, rotated to the horizontal, and enlarged is shown in Fig. 7B. Fig. 8A shows the same point of view, but with the freestream velocity subtracted out. There is some evidence of weak circuitous flow in a few regions and possibly a weak vortex ring near the leading end of the jet. In order to best identify vortices visually, one must subtract the velocity of each potential vortex center. This treatment confirmed the development of weak vortical centers, as seen in Fig. 8A. Apparent flow reversal at the upper left-hand and lower right-hand corners are actually due to the body and wake of the squid, and the flume boundary layer, respectively. Fig. 8B,C shows the tangential and normal velocity profiles of the same jet.
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L/D in squid jets
Fig. 9 shows the
distribution of Lj/D for all 116 squid jets
observed, where Lj is the jet structure length as observed
in the field of view and D is the average jet orifice diameter during
the jet period estimated to be 0.8 cm. Since Lj was
greatly limited by the field of view, the values of
Lj/D in just 21 of the 116 jets were considered
to be good estimates (black bar areas on the histogram). These ranged from 9.0
to 29.5. Lj/D ratios for the remaining 95 jets,
ranging from 4.1 to 32.0, were considered to be underestimates, even gross
underestimates. In 31 of these 95, jet structure clearly extended beyond the
visualized field of view. At the jet angles observed, and assuming the jet
nozzle to be high up on the very upstream edge of the image, the maximum
possible jet lengths that could be measured were 25.1-37.6 cm. This limits
maximum measurable Lj/D ratios to 31.3-46.9.
Since we attempted to capture the jet nozzle of the squid within the field of
view as often as possible, measurable Lj/D ratios
were further limited. This is consistent with the range of
Lj/D in these 31 jets (7.9 to 32.0). During 37
jets, squid exhibited yaw during jetting and therefore the jet was not aligned
with the laser sheet. As mentioned in the Materials and methods, this led to a
visualized jet structure that was shorter than the jet itself.
Lj/D ratios in the jet visualizations affected by
yaw were 5.3-27.9. In at least 24 of the 37 jets, yaw as low as 3° to
14° is sufficient to account for gross underestimates in jet length using
observed jet structure diameters, Dj, of 1.0-2.5 cm.
Recall that Dj is the diameter of the jet flow behind the
squid, not to be confused with jet orifice diameter, D, which is
taken as 0.8 cmthroughout this paper. An additional 27 jets were classified as
underestimates on the basis of both criteria: extension beyond the field of
view and yaw. Lj/D estimates for these jets were
4.1 to 30.9. Despite these limitations, jet structure length does serve to
produce an informative distribution of lower-bound jet length values for the
entire data set. Most significantly, the histogram in
Fig. 9 shows that the more
trustworthy values (Fig. 9,
black bars) characterize squid jets as elongated structures with
Lj/D never less than 9.0. Jet structure lengths
measured when conditions led to underestimates are grouped at the lower end of
the scale. In fact, 84 of the 94 jets with
Lj/D<21 in
Fig. 9 qualify as
underestimates, and both jets with Lj/D<5 were
cases affected by both the limits of the field of view and yaw. Neither the
effect of jet deceleration on jet structure length, nor stretching of the jet
by the motion of the squid through the water, was examined.
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Fig. 10 shows the
distribution of L/D, where L is jet plug length as
determined by the product of average jet velocity and jet period. This ratio
is more appropriate for comparisons of our data to the data of Gharib et al.
(1998). Upper- and lower-bound
jet velocities, jH, and
jL, as defined in the Materials and methods, were
used to determine upper and lower bounds for plug length.
L/D ratios were calculated to be 5.5-31.4 for lower-bound
values (Fig. 10A), and
5.9-61.8 for upper bound (Fig.
10B). A comparison of Figs
9 and
10 shows that the distribution
of the underestimates of Fig. 9
is similar to the distribution of the lower-bound values of
Fig. 10, while the
distribution of the `good estimates' is more similar to the upper bound. On
average, the underestimates of Fig.
9 were increased by 2.9 using the lower-bound jet plug length
calculation and by 13.7 using the upper-bound calculation. For example, the
lowest Lj/D value
(Fig. 9), which was classified
as an underestimate, jumped from 4.1 to between 26.3 and 44.9.
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Jet parameters as a function of swimming speed
Fig. 11 shows the variation
of several parameters of squid jet propulsion with increasing swimming speed.
Trend lines and their equations are included in several of the plots, but are
only meant to aid in the discussion. They should not necessarily be
interpreted as equations that we consider to express the actual relationship
between the variables, nor is it expected that the data should have converged
strongly to a line or curve. Average jet velocity j
(Fig. 11A; `standard value'
from the Materials and methods) increased from 19.9 cm s-1 to 85.8
cm s-1 as swimming speed U increased from 14.2 cm
s-1 to 59.3 cm s-1.
Fig. 11B demonstrates that
scaling jet and swimming velocities by squid mantle length do not
significantly affect the trend observed in
Fig. 11A. Maximum
instantaneous fluid velocities observed within squid jets (not shown) ranged
from 25.6 cm s-1 at a swimming speed of 10.1 cm s-1 to
136.4 cm s-1 at a speed of 25.8 cm s-1.
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Ratios of L/D calculated from average jet velocity
j, orifice diameter D, and jet period
appeared to increase with increasing swimming speed, but there was
considerable variability at speeds between 0.8 and 1.4 Lm
s-1 (Fig. 11C).
Fig. 11D reveals that a
significant degree of that variability arises from variability in jet period.
However, jet period differs from L/D in that it decreases
somewhat with increasing swimming speed. This reveals that the increase in
L/D with increasing swimming speed is not due to the jet
orifice being open longer. Instead, it is due to increasing jet velocities,
j (Fig.
11B). Jet frequency f
(Fig. 11E) also exhibits
increased variability at medium speeds, while values averaged over three or
more cycles for the same squid and speeds, favg, reveal an
interesting trend. Average frequency favg lies between 0.8
and 1.2 Hz and is highest at the lowest and highest swimming speeds observed.
A parabolic fit to favg predicts a minimum average jet
frequency at about 0.9 Lm s-1.
Anderson and DeMont (2000)
reported an increase in jet frequency as swimming speed increased from 1.0 to
1.7 Lm s-1. Yet, their data suggested that the
increase in frequency was achieved by varying primarily the refill period.
Fig. 11D suggests, however,
that changes in jet period are also responsible for the trend in jet
frequency. In fact, we found that trends in both jet period and refill period
as percentages of the total locomotive period were relatively constant near
31% and 46%, respectively, for the swimming speeds observed, with significant
variability at medium speeds. The remaining 23% of the cycle represents the
period during which the jet was opening and closing. Therefore, shorter
locomotive periods at high and low swimming speeds were achieved by shortening
both the refill and jet periods. Interestingly, for the medium speeds, at
which considerable variability in jet frequency was observed, shorter
locomotive periods were achieved by decreasing the percentage of the
locomotive period taken up by refill while increasing the fraction taken up by
opening, closing and jetting.
Jet angle ß decreased with increasing swimming speed (Figs
5,
11F). This is undoubtedly
related to the maintenance of a relatively constant upward component of jet
thrust to counter the squid's constant negative buoyancy. The higher jet
velocities j at higher swimming speeds produce a
greater thrust along the axis of the jet. If the jet angle were not decreased
the squid would move up in the water column. Negative buoyancy has been
measured in several squid, including Loligo forbesi
(Denton, 1961) and
Ommastrephes sagittatus (Zuev,
1963). At higher speeds the squid may also be capitalizing on
increased lift to support its weight, further decreasing the jet angle
necessary for maintaining constant vertical position. Zuev
(1965) investigated the squid
body as an airfoil and observed lift in experimental models. Such factors
suggest that jet angle should asymptotically approach zero with increased
swimming speed, and indeed, a natural logarithmic curve fit our jet angle data
better than a straight line. O'Dor
(1988) and Bartol et al.
(2001b) have observed a
decrease in angle of attack of the body with increasing swimming speed in
steadily swimming squid, further suggesting the importance of lift in the
balance of vertical forces.
Propulsive efficiency
Fig. 12A shows slip as a
function of swimming speed based on the average jet velocity
j, and the component of the flume flow in the
direction of the jet, Ucosß. A broken line shows the trend if
jet angle ß is ignored. The trend-lines for slip calculated with and
without considering jet angle vary less as speed increases
(Fig. 12A), since jet angle
decreases with increased swimming speed
(Fig. 11F). The average
difference between the two calculations of slip for our data was 10%. Either
way, the data suggest a decrease in slip with increasing swimming speed,
asymptotically approaching 0, which is a trend common to plots of slip in
vehicles driven by jets and propellers. It reveals a relative decrease in the
excess kinetic energy left in the wake with increased swimming speed, and
therefore a higher propulsive efficiency. Propulsive efficiency during jetting
j from Eq. 3 (Fig.
12B) reaches relatively high values, even at medium speeds, and
then levels off. The average efficiency is 86% for speeds above 0.65
Lm s-1 and 93% for speeds above 1.6
Lm s-1, with efficiencies for a handful of jet
events reaching 95-97% at speeds above 0.9 Lm
s-1. These values may be slightly high since the standard jet
velocity used in the calculation of efficiency may underestimate true jet
velocity. Fig. 12C confirms
that propulsive efficiency calculated from Eq. 3 (broken curve) matches very
nicely with that calculated from estimates of total and excess kinetic energy
of the jet.
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Whole-cycle hydrodynamic efficiencies wc (not shown),
calculated using Eq. 4, were much lower than jet propulsive efficiency, as
expected, due to the high cost of refill. Average values ranged from 42 to
49%. Anderson and DeMont (2000)
report a range of 34-48%. However, taking their observed swimming speeds, jet
velocities, jet angles and applying the same estimations concerning refill
area and period that we used (see Materials and methods), one obtains a range
of 38-44%. Their lower-bound value of 34% was obtained because they purposely
set the upper-bound refill velocity equal to average jet velocity. This is
certainly too high, but served to predict a trustworthy lower bound for whole
cycle hydrodynamic efficiency. Our observations during the refill period,
however, suggest an upper-bound refill velocity closer to 0.5 times average
jet velocity (see Materials and methods).
Jet thrust
Fig. 12D shows average jet
thrust in squid as a function of swimming speed calculated by two methods: (1)
from a three-dimensional approximation of the change in momentum per time
between successive visualizations of jets, and (2) using the upper-bound jet
velocity in the simplified steady equation for jet thrust along the jet axis
(i.e. 
, where is the
density of seawater, A is the cross-sectional area of the jet,
uj is the magnitude of the jet velocity). Both methods can
only be considered approximations in comparison to jet thrust determined from
the actual jet velocity at the jet orifice and the orifice area as functions
of time, as in Anderson and DeMont
(2000). Nevertheless, their
average jet thrust of 0.030 Ndetermined in a squid swimming 1.0
Lm s-1, including unsteady effects, falls
within the range of our estimates from velocity and momentum of 0.017 to 0.042
N. This agrees well with the necessary average jet thrust of approximately
0.030 N determined by Anderson et al.
(2001c) from the acceleration
of and drag on similarly sized squid during jetting at the same swimming speed
(see their Fig. 10). We used
our upper-bound jet velocity in Fig.
12D because it was closer to the more reliable measurements of
Anderson et al. (2001c).
Jets from pipes with background flow
So far, we have focused on jets of squid without much mention of the impact
of background flow. Flow visualization of jets emitted from pipes revealed the
important role of background flow in the development of the fluid structure of
squid jets. As described earlier, our pipe jet results for jets emitted into
still water (U=0) matched the findings of Gharib et al.
(1998)
(Fig. 2). Fig. 2A shows the predicted
formation of a single vortex ring when L/D=4.3. By contrast,
Fig. 2B shows that a trail of
vorticity follows the leading vortex ring when L/D=16. In
both cases, piston velocity up was 5 cm s-1,
but since, U=0, the ratio of background flow velocity to piston
velocity, U/up, was zero.
Fig. 13 illustrates the impact
of increasing levels of background flow on jet structure. The ratios of
background flow to piston velocity shown are 0.5, 1.0 and 2.0, respectively.
Even at the lowest level of non-zero background flow
(U/up=0.5), jet structure
(Fig. 13A,B) was different
from the still-water case (Fig.
2A,B). When L/D=4.3
(Fig. 13A) there was no longer
a single, well-defined vortex ring as in still water
(Fig. 2A). Instead, jet
vorticity was spread over a distance of about 10 cm and suggested a deformed
leading vortex ring with some trailing vorticity. At L/D=16
(Fig. 13B), the leading
vortical structure was also deformed and slightly less prominent than observed
in still water (Fig. 2B). Only
slight remnants of vorticity originating from the outer boundary layer and of
magnitude comparable to jet vorticity were present around the jet flow
(Fig. 13A,B). At the next
level, in which background flow velocity was equal to piston velocity,
U/up=1.0
(Fig. 13C,D), the effects
observed in Fig. 13A,B were
much more obvious. For the case of L/D=4.3
(Fig. 13C), all of the jet
vorticity was now spread out into an elongated structure. There is little to
no suggestion of a prominent leading vortex ring as in Figs
2A and
13A. When
L/D=16 (Fig.
13D) the prominent leading vortex, as seen in still water
(Fig. 2B), is diminished, or
not observed. More significant remnants of the outer boundary layer are
visible at U/up=1.0 than at
U/up=0.5. Finally, when background flow velocity
is twice the piston velocity, U/up=2.0
(Fig. 13E,F), the outer
boundary layer vorticity dominates and there is essentially no comparable jet
vorticity present. Although this level of background flow is not assumed to be
generally applicable to squid, it helps to illustrate a continuum in the
relative levels of dominance of the inner and outer boundary layer vorticities
as the level of background flow, U/up, is
increased.
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